Definitions

Given a setSS, let a point in SS be an element of SS, and let a set in SS be a subset of SS. Given a relation∈∘\stackrel{\circ}\in between points in SS and sets in SS, say that the set UU is a neighbourhood (or ∈∘\stackrel{\circ}\in-neighbourhood to be precise) of xx if x∈∘Ux \stackrel{\circ}\in U (which may also be written x∈U∘x \in \stackrel{\circ}U).

A pretopology (or pretopological structure) on SS is such a relation ∈∘\stackrel{\circ}\in that satisfies these properties:

Centred: If UU is a neighbourhood of xx, then xx belongs to UU:

x∈∘U⇒x∈U. x \stackrel{\circ}\in U \;\Rightarrow\; x \in U .

Nontrivial: Every point xx has a neighbourhood. In light of (4), the entire space is a neighbourhood of xx:

x∈∘S. x \stackrel{\circ}\in S .

(Some references leave this out, but that seems to be an error.)

Directed: If UU and VV are neighbourhoods of xx, then so is some set contained in their intersection. In the light of (4), it follows that their intersection is itself a neighbourhood:

In other words, the collection of neighbourhoods of xx must be a filter that is refined by the free ultrafilter at xx. This filter is called the neighbourhood filter of xx.

A pretopology can also be given by a base or subbase. A base for a pretopology is any relation that satisfies (1–3), using the first version for each of (2,3); a subbase is any relation that satisfies (1). (It would not really be appropriate to use the symbol ‘∈∘\stackrel{\circ}\in‘ for a mere base or subbase; you'd probably want to think of it as a family of sets indexed by the points, and use the term ‘basic neighourhood’ or ‘subbasic neighbourhood’.) You get a base (in fact one satisfying the stronger versions of 2,3) from a subbase by closing under finitary intersections; you get a pretopology from a base by taking supsersets. (Really, this is just a special case of considering a base or subbase of a filter.)

A pretopological space is a set equipped with a pretopological structure.

A continuous map from a pretopological space SS to a pretopological space TT is a functionff from SS to TT such that:

For every point xx in SS, if UU is a neighbourhood of f(x)f(x) in TT, then the preimagef*(U)f^*(U) is a neighbourhood of xx in SS.

In this way, pretopological spaces and continuous maps form a categoryPreTopPre Top.

Convergence structure

If FF is a filter on a pretopological space SS, then FFconverges to a point xx (written F→xF \to x) if FF refines (contains) the neighbourhood filter of xx.

This relation satisfies the following properties:

Centred: The free ultrafilter at xx (the collection of all sets that xx belongs to) converges to xx:

{A|x∈A}→x. \{ A \;|\; x \in A \} \to x .

Isotone: If FF converges to xx and GG refines FF, then GG converges to xx:

In fact, we can recover the pretopological structure from the convergence structure as follows: x∈∘Ux \stackrel{\circ}\in U if and only if UU belongs to every filter that converges to xx. In other words, that intersection that appears in the infinite filtration condition is the neighbourhood filter of xx. Furthermore, this definition assigns a pretopological structure to any convergence space satsifying the conditions above, and a map between pretopological spaces is continuous if and only if it is continuous as a map between convergence spaces. Thus, we can define a pretopological space as an infinitely directed convergence space, making PreTopPre Top a full subcategory of the category ConvConv of convergence spaces.

Actually, we can do more. The definition of U∘\stackrel{\circ}U from the convergence structure assigns a pretopological structure to any convergence space, although in general this pretopology defines a weaker notion of convergence (more filters converge to more points). Thus, PreTopPre Top is also a reflective subcategory of ConvConv.

Every pretopological convergence satisfies the star property, so that PreTopPre Top is a full reflective subcategory of the category PsTopPs Top of pseudotopological spaces.

Examples

Every topological space is a pretopological space, using the usual definition of (not necessarily open) neighbourhood: x∈∘Ux \stackrel{\circ}\in U if there exists some open set GG such that x∈Gx \in G and G⊆UG \subseteq U. Also, a map between topological spaces is continuous if and only if it's continuous as a map between pretopological spaces. In this way, the category Top of topological spaces becomes a full subcategory of PreTopPre Top.

In fact, we can easily characterise the topological pretopologies, allowing us to define a topological space as a pretopological space satisfying this axiom:

If UU is a set, then let U∘\stackrel{\circ}U be the set of all points that UU is a neighbourhood of. Then U∘\stackrel{\circ}U is a neighbourhood of each of its members. That is,

In the terms defined below, a topological space is a pretopological space in which every preinterior is open.

Here is an example of a nontopological pretopological space, although admittedly it is a bit artificial. (This is based on Section 15.6 of HAF.) Consider a metric spaceSS; according to the usual pretopology on SS, UU is a neighbourhood of xx if there is a positive number ϵ\epsilon such that UU contains the ball {y|d(x,y)<ϵ} \{ y \;|\; d(x,y) \lt \epsilon \} . Now given a natural numbernn, we will give SnS^n the plus pretopology: UU is a neighbourhood of x⇀=(x1,…,xn)\vec{x} = (x_1,\ldots,x_n) if there is a positive number ϵ\epsilon such that UU contains the l0l^0-ball {y⇀|infid(xi,yi)<ϵ} \{ \vec{y} \;|\; \inf_i d(x_i,y_i) \lt \epsilon \} . (If SS is a line and n=2n = 2, then this neighbourhood is a plus sign ‘+’ with (x1,x2)(x_1,x_2) at the centre and cross bars of length 2ϵ2 \epsilon.) Then SnS^n is a pretopological space, but it is topological only if n≤1n \leq 1 or SS is a subsingleton.

This example can probably be generalised to a uniform spaceSS. Possibly there is some interesting universal property of this ‘plus product’, although it seems to go from Unif×UnifUnif \times Unif to PreTopPre Top, so maybe we need to work in a different category. (There is a notion of uniform convergence space that generalises uniform spaces much like convergence spaces generalise topological spaces; perhaps the plus product takes place there.)

Topological structure

Fix a pretopological space SS.

The preinterior of a set AA is the set A∘\stackrel{\circ}A or A∘A^\circ of all points that AA is a neighbourhood of:

A set AA is open if it equals its preinterior. The interiorInt(A)Int(A) of AA is the union of all of the open sets contained in AA. Note that we can immediately recover the pretopological structure from the preinterior operation (but not from the interior operation nor from the class of all open sets).

Similarly, the preclosure of AA is the set A¯\bar{A} of all points that AA meets every neighbourhood of:

A set AA is closed if it equals its preclosure. The closureCl(A)Cl(A) of AA is the intersection of all of the closed sets containing AA. Again, we can recover the pretopological structure from the preclosure operation; x∈∘Ux \stackrel{\circ}\in U iff UU meets every set AA such that x∈A¯x \in \bar{A}. (This result seems to require excluded middle.)

(Warning: not all references use these terms in the same way. This terminology is based on the premise that a closure should be closed.)

The duality between (pre)interiors and open sets on the one hand and (pre)closures and closed sets on the other hand is (at least if you assume excluded middle) just what you would expect: the (pre)interior of a complement is the complement of the (pre)closure, and a set is open if and only if its complement is closed. However, a preinterior is generally not open but larger than an interior; similarly, a preclosure is generally not closed but smaller than a closure. The situation looks like this:

In many cases this iteration stabilizes after finitely many terms. The plus power SnS^n seems to stabilise after nn iterations. And in a topological space, of course, it only takes one step.

In general, however, there can be transfinitely many terms in these sequences. For example, let Ω\Omega be any ordinal number (thought of as the well-ordered set of all smaller ordinal numbers) with the following pretopology:

Note that an interior is open, and a closure is closed. Indeed, the open sets in SS form a topological structure on SS, giving the usual meanings of interior, closure, and closed set. This topological structure does not (in general) give the original pretopology on SS; instead, this makes TopTop a reflective subcategory of PreTopPre Top.

In the definition of pretopology, the neighbourhoods of each point may be given completely independently of any other point. So the notion of topological space may also be seen as requiring some coherence between the neighbourhoods of nearby points.